σ Standard Deviation Calculator
Calculate mean, variance, and standard deviation for population or sample data with a full statistical breakdown.
Enter Your Data
Enter numbers separated by commas, spaces, or new lines
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Mean (x̄)
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Std Deviation
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Variance
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Median
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Range
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Count (n)
Formulas
- Population σ = √[Σ(xᵢ−μ)² / N]
- Sample s = √[Σ(xᵢ−x̄)² / (n−1)] — Bessel's correction
- Variance = σ² or s²
- Use sample (s) when data is a subset of a larger population
How the Standard Deviation Calculator Works
How Standard Deviation Works
Standard deviation measures how spread out data points are from the average (mean). A low standard deviation means data points are close to the mean, while a high standard deviation means they are spread out.
Mathematical Formulas:
Population Standard Deviation:
σ = √[Σ(xi – μ)² / N]
Use when you have the entire population
σ = √[Σ(xi – μ)² / N]
Use when you have the entire population
Sample Standard Deviation:
s = √[Σ(xi – x̄)² / (n-1)]
Use when you have a sample from a larger population
s = √[Σ(xi – x̄)² / (n-1)]
Use when you have a sample from a larger population
Variance:
Variance = (Standard Deviation)²
Average of squared differences from mean
Variance = (Standard Deviation)²
Average of squared differences from mean
Real-World Applications:
- Quality Control: Manufacturing tolerance, product consistency
- Finance: Investment risk analysis, portfolio volatility
- Education: Test score analysis, grade distribution
- Healthcare: Patient vital signs, clinical trial analysis
- Research: Experimental data analysis, survey results
Usage Recommendations:
- Use population standard deviation when you have complete data (entire population)
- Use sample standard deviation when working with a subset of data
- Remove outliers before calculation if they represent errors, not true variation
- Standard deviation is most meaningful for normally distributed data
- Consider the 68-95-99.7 rule: ~68% of data falls within 1σ of the mean